Let G be a semi-simple algebraic group over an algebraically closed
field of characteristic zero, and TÌBÌG a maximal torus
and Borel subgroup respectively. We will state a general result about
the set of smooth points of a closed subvariety YÌG/B stable
under T which allows us to characterize the smooth points of a
Schubert variety X in G/B. In particular, a Schubert variety X is
nonsingular iff its reduced tangent cones are all linear. The reason
this result is nontrivial is that the reduced tangent cone of an X at
a point x almost never spans the tangent space to X at x. A
corollary of this is a well known unpublished result of Dale Peterson
that if G is simply laced, then every rationally smooth Schubert
variety is in fact already smooth. (Joint work with Jochen Kuttler).

There are only finitely many adjoint nilpotent orbits O in a simple
real Lie algebra g0. The closure of O is the union of O and some
orbits of smaller dimension. I shall discuss the problem of deciding
which orbits are contained in the closure of O. An approach to this
problem via the Kostant-Sekiguchi correspondence will be described.
Surprisingly, this problem is still open when g0 = so(p,q). Some
partial results for this case were obtained in a joint work with
J. Sekiguchi and N. Lemire. I shall also present the results, obtained
so far, in the case when g0 is a noncompact real form of a simple
exceptional complex Lie algebra.

Suppose P®X is a holomorphic principal bundle and a Lie
group G acts as a group of holomorphic automorphisms on the bundle P®X. We will show that if there exists a holomorphic action
of the complexification G\mathbb C of G on X, then this lifts
to a holomorphic action of G\mathbb C on the bundle P®X. Applications of this will also be presented.

The pro-algebraic hull A(G) of a discrete group G is the inverse
limit of the Zariski closures of all finite dimensional complex
representations of G. We consider only finitely generated G whose
representations separate the elements of G (in the finitely generated
case, this is the same as residually finite). Examples are presented
where the (pro)unipotent radical of A(G) is infinite dimensional and
the reductive quotient finite dimensional, and where the prounipotent
radical is trivial but A(G) is infinite dimensional and the
occurrence of simple factors in A(G) of given dimension is bounded.
(Joint work with H. Bass, A. Lubotzky, S. Mozes)

By work of De Concini, Procesi and Springer, there exists a cell
decomposition of X = [`(G)]:

X = \bigsqcupaÎW ×WCa

where W is the Weyl group and Ca@Kma for
some ma³ 0. Each Ca is B×B-invariant.
Furthermore, they calculate each ma, thereby yielding a
formula for the Betti numbers of X.

We take a more direct approach. First we identify the set of B×B-orbits combinatorically in terms of (W, f, D), and then we
define each Ca explicitly as a union of B×B-orbits.
Our approach also determines a cell decomposition of each G×G-orbit closure XJÍX, JÍD.

If G® GL(V) is a representation of the finite group G
on the vector space V over a local field, we will describe the
symmetric bilinear forms on V which are G-invariant, and count the
number of orthogonally equivalent orthogonal representations for each
of these invariant forms. Partial results for global fields will also
be given.

Suppose that V is a finite dimensional vector space over the field
k and that G is a finite subgroup of GL(V). There is an
induced action of G on the ring of regular functions k[V].
The ring of invariants, k[V]G, is the finitely generated subring
consisting of those functions fixed by the action of G. If the order
of G is a unit in k, then k[V]G is reasonably well
understood and there are fairly effective algorithms for constructing a
minimal generating set. This is not the case if the characteristic of
k divides the order of G.

Then there exists a normal subgroup, N, with G/N isomorphic to the
cyclic group of order p, Z/p. The ring of invariants is given by

k[V]G = (k[V]N)G/N = (k[V]N)Z/p.

Therefore it is possible, in principle, to construct a generating set
inductively. In other words, to climb the ladder from k[V] to
k[V]G. Each step up the ladder requires the computation of the
Z/p-invariants of a finitely generated graded algebra. Computing
Z/p-invariants can be difficult and this approach is not practical
for an arbitrary p-group. There are, however, cases were the ladder
approach works well.

I intend to discuss the elements required to make the ladder approach
work. This involves some interesting group cohomology calculations. I
will also describe a family of representations for which the ladder
approach has proven to be more effective than any other currently known
method. (Joint work with Dr R. James Shank (Institute of Mathematics
& Statistics, University of Kent at Canterbury).